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Counter-propagating wave patterns in a swarm model with memory.

Angelika Manhart1

  • 1Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY, 10012, USA. angelika.manhart@cims.nyu.edu.

Journal of Mathematical Biology
|August 30, 2018
PubMed
Summary
This summary is machine-generated.

We modeled motile organism movement using hyperbolic transport-reaction equations, introducing memory effects for turning individuals. Our study explicitly constructs counter-propagating traveling waves observed in bacterial colonies.

Keywords:
Age-structured equationsHyperbolic equationsMyxobacteriaPattern formationTraveling wavesViscous limitWave formation

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Area of Science:

  • Mathematical Biology
  • Biophysics
  • Partial Differential Equations

Background:

  • Hyperbolic transport-reaction equations model motile organism movement.
  • Age-structuring is crucial for understanding population dynamics.
  • Bacterial colonies exhibit complex spatial patterns like traveling waves.

Purpose of the Study:

  • To develop a mathematical model for age-structured motile organisms with memory.
  • To analyze the emergence of counter-propagating traveling waves.
  • To investigate conditions for wave formation and stability.

Main Methods:

  • Formulation of a system of four coupled hyperbolic transport-reaction equations.
  • Inclusion of a memory effect based on time since last behavioral reversal.
  • Explicit construction and mathematical characterization of traveling wave solutions.
  • Stability analysis of wave patterns and homogeneous solutions.

Main Results:

  • Successfully constructed and characterized counter-propagating traveling waves.
  • Identified conditions leading to the formation of these wave patterns.
  • Revealed the existence of pulsating-in-time spatially constant solutions.
  • Demonstrated the relevance of memory effects in biological transport.

Conclusions:

  • The model captures emergent spatial patterns in bacterial colonies.
  • Memory effects play a significant role in the dynamics of motile populations.
  • The study provides a framework for analyzing complex behaviors in biological systems.